Master's Degree Thesis ISRN: BTH-AMT-EX--2012/D-24--SE

Supervisors: Ansel Berghuvud, BTH

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden

2012

Hamza Zeidan Nasser Shatha Hameed Hajwal

Fluid Structure Interaction Modeling of the Dynamic of a Semi

Submerged Buoy

_________________________________________ Master’s Degree ThesisISBN

[1]

Fluid Structure Interaction

Modeling of the Dynamic of a Semi Submerged Buoy

Hamza Zeidan Nasser

Shatha Hameed Hajwal

Department of Mechanical EngineeringBlekinge Institute of Technology

Karlskrona Sweden2012

_________________________________________________________________________Supervisor: Ansel Berghuvud, Ph.D.Mech. Eng.

[2]

Hamza Zeidan Nasser

Shatha Hameed Hajwal

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona Sweden2012

Thesis submitted for completion of Master of Science in Mechanical Engineering with

Emphasis on Structural mechanics at the Department of Mechanical Engineering,

Blekinge Institute of Technology, Karlskrona, Sweden.

Abstract:

This thesis presents a study of buoy systems for wave’s energy by focusing on the

development of a model in which modeling of a wave energy conversation is in

operation. Throughout the thesis, the buoyancy and motion of the submerged body has

been used to describe the wave-buoy interaction. The mathematical model for an

investigating buoyancy and the dynamic heave response of this buoy under the two

different load cases be considered as a single degree of freedom, which have natural

characteristic, mass (M), stiffness (K) and damper (C) of the oscillating buoy which

were estimated from the numerical method, because it is very important in the ocean

wave stage and essential for the design of offshore installation. By using the Finite

Element analysis in the software COMSOL Multiphysics 4.2.a, the fluid structure

interactions established on two dimensional potential flow theories and linear are

simulated and modeled. Then compared the numerical results which are obtained by

transient harmonic oscillations with these of different geometric.

Comparison the analytical results and experimental system lent validity to the numerical

models.

Keywords

COMSOL Multiphysics, wave energy converter, Dynamic characteristic, Submerged

body, Finite Element method (FE), SDOF, ALE,

[3]

Acknowledgement:

The work presented in this thesis has been carried out at the Department of Mechanical

Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, under the supervisor

of Dr. Ansel Berghuvud.

The work was performed with deep gratitude the continuing support, the valuable

comments and assistance received from the supervisor, Dr. Ansel Berghuvud, for being

there to encouragement, guidance and provide answers to all questions we had during

this work to make it a successful and extremely meaningful.

Finally, we express our deepest gratitude goes to our families for their continued care,

support and encouragement throughout our life.

Karlskrona, 2012 October

Hamza Zeidan NasserShatha Hameed Hajwal

[4]

Table of Contents

1. Introduction ..................................................................................................................................... 8

1.1 Background ................................................................................................................................. 8

1.2 Aim and Objective....................................................................................................................... 9

1.3 Scope ......................................................................................................................................... 10

2 Theoretical and Basic Notations.................................................................................................... 11

2.1 Scope Conceptual ...................................................................................................................... 11

2.2 Buoy Dynamics ......................................................................................................................... 11

2.2.1 Hydrodynamic Issues ............................................................................................................ 12

2.2.2 Boundary Conditions............................................................................................................. 13

2.2.3 Added Mass and Damping .................................................................................................... 15

2.3 FSI Consideration...................................................................................................................... 17

2.3.1 Laminar, Two-Phase Flow, Phase Field (tpf) Physics Application....................................... 17

2.3.3 Arbitrary Lagrangain-Eulerain (ALE) Physics Application.................................................. 20

2.4 Finite Element Analysis ............................................................................................................ 21

2.4.3 Practical Application ............................................................................................................. 22

2.5 Theoretical Model ..................................................................................................................... 22

2.5.1 Single Degree of Freedom Model ......................................................................................... 23

2.5.2 Mathematical Model.............................................................................................................. 26

3 Study Methodology ....................................................................................................................... 28

3.1 Modeling ................................................................................................................................... 28

3.1.1 Model Specification .............................................................................................................. 28

3.1.2 Boundary conditions.............................................................................................................. 29

3.1.2.1 Moving Mesh Boundary Condition....................................................................................... 29

3.1.2.2 Solid Domain Boundary Condition ....................................................................................... 29

3.1.2.3 Water and Air Domain Boundary Condition......................................................................... 32

3.2 Model Meshing.......................................................................................................................... 32

3.3 Model Solver ............................................................................................................................. 34

4 Result and Discussion ................................................................................................................... 35

4.1 Results ....................................................................................................................................... 35

4.2 Parameters Estimation ............................................................................................................... 41

5. Conclusion..................................................................................................................................... 42

[5]

Future work ........................................................................................................................................... 42

6. Reference....................................................................................................................................... 43

Appendix 1 ............................................................................................................................................ 46

c. Study.......................................................................................................................................... 66

[6]

Notations

, Added Damping [Ns/m]

Damping of a System [Ns/m]

Modulus of Elasticity (Young’s Modulus) [Pa]

User defined Free Energy [J/m ]

g Gravitational Force [9.81 N/m ]

j Imaginary Unit [j= 1]

Buoyancy Stiffness [N/m]

k Stiffness of the System [N/m]

Added Mass [kg]

Mass of the Floating Buoy [kg]

m Mass of the System [kg]

P Pressure [N/m ]

Atmospheric Pressure [N/m ]

Complex Variable

Poisson’s ratio

Mobility Tuning Parameter

Mobility [m . s/kg]

Damping Ratio [ = . ]

[7]

Dynamic Viscosity [Ns/m ]

Density of Fluid [kg/m ]

Surface Tension Coefficient in Fluid Dynamic

Angular Frequency [rad/s]

Contact Angle [rad]

Dependent Variable, the Phase field Variable

Dependent Variable, the Phase field Help Variable

Mean Free Path [m]

Parameter Controlling Interface Thickness, Stress Tensor in Structural

Mechanics

Relative Permittivity

Abbreviations

FSI Fluid Structure Interaction

FEM Finite Element Method

ALE Arbitrary Lagrangian – Eulerian

SDOF Single Degree of Freedom System

CFD Computational Fluid Dynamics

CSD Computational Structural Dynamics

[8]

1. Introduction

The present moment, there is an intense interest to be working in the field of wave energy

research. Due to increasing energy demand in the world at an alarming rate and

production of electricity from renewable energy sources which is becoming an important

in the idea of the development of this technology to become a major player in the energy

industry. Many countries are setting ambitious targets in this topic. This growing

enthusiasm can be seen in several forms, such as attendance of wave energy conferences,

academic research funding and political support. With those support of mechanisms, the

wave energy technologies has development. Therefore it is very important to find ways to

harness energy from sustainable and clean sources.

Wave energy has been under investigation by researchers for decades and continues to be

challenging problem. There are many forms of renewable energy system around the

world, such as wind mills, ocean energy, tidal and offshore wind, nuclear power and solar

power.

Robertson [In the late 1860’s] discovered the first wave power boat by harnessing the power

of the wave, also inventors and engineers have focused on wave powered technologies over

the last century. Their studies are important to understand this resource to extract its energy to

produce electricity.

1.1 Background

This thesis “Fluid Structure Interaction Modeling of the Dynamic of a Semi Submerged

Buoy” is an extended of the previous work, in which the floating buoy has a stable mass

[9]. The mechanical of single degree of freedom model was present the hydrodynamic

parameters which was estimated from the numerical model. In this work, COMSOL

Multiphysics software was used to simplify the Finite Element (FE) and numerical single

degree of freedom (SDOF) basic on the knowledge of the dynamic of the submerged

buoy. The basic for the linearized forces which lets superposition of exciting waves,

added damping, added mass, and restoring force was established by the linear theory [9,

10, 11].

[9]

To formulate the mathematical model of the buoy, assumptions and approximation were

involved; therefore the model was establish and assessed by many ways of validation

against its analytical model. A complex phenomenon which happened in Fluid Structure

Interaction (FSI) analysis was the coupling between the structural dynamics and the fluid

dynamics. Bothe of the computational structural dynamics (CSD) and computational fluid

dynamics (CFD) must couple through the governing equations of interaction between the

floating buoy and the water. First must be created the governing equations for the water

in the absence of the floating. Stress-Strain application and strain-displacement from

structural mechanics were used to define the floating buoy movement. The model

equations of the fluid were described by generalized Navier Stokes equation which

solved both of velocity and pressure. The traction and the displacement compatibility

should be specified to resolve the coupling of the buoy-fluid interface in order to ensure

the equilibrium.

1.2 Aim and Objective

The aim of this work is to model the dynamic characteristic parameter of the submerged

buoy with the variables m (Mass), c (Damping) and k (Stiffness). On the basic of this

linear theory, single degree of freedom (SDOF) mathematical models of the floating buoy

are formed to demonstrate the motion of oscillation. The main section of its objective to

achieve a mathematical representation of the semi submerged cubic buoy which was

formed upon single degree of freedom with it is characteristic, mass (M), stiffness (K)

and damper (C) of the oscillating buoy which were estimated from the numerical method

solved by carrying out the Finite Element analysis in the software COMSOL

Multiphysics 4.2.a. Comparison with numerical models lent validity to the experimental

and analytical results which are investigated from the work.

[10]

1.3 Scope

The scope of this thesis is to present the model of the dynamic analysis of the buoy

oscillating in finite water depth to a single mode of the oscillation, and dealing with the

coupling effect of the submerged floating buoy. The numerical models and the

experimental investigation are present here with two geometries with the presumption of

having different masses. The added mass and added damping coefficients are obtain from

determining the dynamic pressure and energy relations, but the buoyancy stiffness was

separately assessed in relation to hydrostatic pressure. On the fundamental of linear

theory, the single degree of freedom model of the planar oscillating buoy is design to

demonstrate the vertical oscillation in the water of constant depth. From the motion of a

submerged body and the modeling of the buoyancy, SDOF model is validated against the

experimental results and analytical model [9].

[11]

2 Theoretical and Basic Notations

2.1 Scope Conceptual

This thesis is an extension to find a method to predict the floating buoy motion and find a

simulation model. It is important to know the wave of the force coefficients, the added

damping coefficients and the added mass coefficients to predict that motion of the

floating buoy which are subjected ocean waves. By the SDOF system, the heave motion

of the floating cube buoy in fluid domain with the constant depth water can be modeled.

The computational fluid-structure interaction (FSI) analysis was detailed in relation with

software COMSOL Multiphsics. With the accommodation of the Arbitrary Lagrangian-

Eulerian (ALE), the problems due to exact delineation and deformations form the general

moving mesh method are find out. The FSI predefined Multiphysics included a

continuous application mode Stress-Strain, Laminar, Two-Phase Flow, Phase Field (tpf),

and Moving Mesh (ale). The problems which are dealing with (FSI) requires coupling of

Computational Structural (CSD) and Computational Fluid Dynamics (CFD). In several

industrial applications, (FSI) plays a very significant role such as biomedical, nuclear

engineering, material processing and automotive etc. The coupling between the structure

and the fluid requires sharing of the boundary condition between them. The coupled

fluid-structure governing equations can be resolved by using COMSOL Multiphysics

solver [12].

2.2 Buoy Dynamics

The figure (2.1) below shows the description of the motion of a submerged buoy when

been excited by surface wave; due to that excitation the buoy experiences six degrees of

motion; two for each direction rotation and translation, as:

X direction: roll as rotation and surge as translation

Y direction: Pitch as rotation and sway as translation

Z direction: Yaw as rotation and heave as translation

[12]

Figure2.1. Six motion for the buoy oscillating in the water

Due to the heave motion is necessary with regard to the oscillations of the buoy; the buoy

motion will investigate only in the z-direction.

2.2.1 Hydrodynamic Issues

The assumptions taken for the governing equation are:

1. The oscillation is only in heave direction.

2. The friction losses and viscosity been ignored.

Hydrodynamic loads are the classifier of the hydrodynamic problems [13]:

I. The incident regular waves restrained the structure from oscillation while

the excitation load waves cause the oscillation.

Sway

Surge

Yaw

Roll

Pitch

Heave

Z-direction

Y-directionX-direction

[13]

II. The hydrodynamics loads are defined as added damping, added mass and

restoration terms.

The added damping and mass where due to the FSI states, where the added mass where

described by a matrix which representing the modeling of interaction between the

structure and the surround fluids, which that surround fluid increase the effective mass of

the structure where the added mass is equal to that fluid displaced mass, taking in

consideration that the added mass term assume for an ideal fluid independent on the

viscosity. [5]

2.2.2 Boundary Conditions

Pressure fields in the fluid are caused due to the oscillation of the structure which would

form outgoing waves, under the absence of incident waves where the oscillation assumed

to be formed due to that forced harmonic heave motion of the floating buoy.

Figure2.2 Boundaries of the floating buoy

With surroundings

Z-direction

X-direction

[14]

Where:

: The boundary condition of the linearized free surface.

: The buoy surface.

The boundary conditions ( ) were specified as [14,17,19,20]:

1. The boundary condition free-surface are consider to be linear

= (2.1)

Where [k] represent the number of wave, which has the equation of = [20]. The

water pressure on the free surface was assumed to be equal to the atmospheric pressure

to satisfy the dynamic free-surface condition.

= (2.2)

Velocity potential which is satisfying the Laplace equation for the fluid domain is

specified by determining the initial conditions and boundary conditions.

2. At the surface, the normal vector for the forced heave velocity and the fluid

velocity are required to be equal, figure (2.2).

= (2.3)

= 0 = (2.4)

Where:

u: The velocity of the buoy.

:The normal vector component of velocity of this buoy.

n: The normal vector for the wetted surface directed in the z-direction.

[15]

3. On the bottom at depth h, the normal vector of the fluid velocity is equal to

zero considering the fluid as an ideal.

= 0 (2.5)

4. At infinite distance, the radiations of the leaving wave were considered to

be as:

± 0 ± (2.6)

2.2.3 Added Mass and Damping

By determining the dynamic pressure after solving the potential velocity, the added mass

can be obtained. Ignoring the hydrostatics pressure and integrating the remaining pressure

covered the buoy; the vertical force should be as following [14]:

= (2.7)

Where:

Fz: The force which is obtained from integration of the linear pressure.

ma: The added mass in the heave mode.

Z: The displacement in the heave direction.

Ca: The coefficient of the heave damping.

Damping coefficient can be obtained from the energy relations. Since the damping is

related to the energy, so the total energy written as summation of potential and kinetic

energy [14].

( ) = (1

2+ ) (2.8)

The derivative of the total energy over the time

[16]

dE(T)

dt= (

t n

p p+

t)U )ds (2.9)

Where:

E: The total energy for the fluid volume [ ].

: The volume integration.

Sur: The boundary surface of the fluid volume.

: The derivation on along the unit normal vector with the surface boundary.

P0: The atmospheric pressure.

Un: The normal component of the mean velocity for the surface boundary.

With respect to the above boundary conditions for this problem, which consisted of the

wetted buoy surface Surb, the Sur is bounded from - the derivative of the total

energy should be as following:

= ( ) (2.10)

( ) = ( ) (2.11)

( ) = + + (2.12)

The two terms [ ] and [ ] originated due to the hydrostatic pressure. Where the

[ ] represented the change in the buoyancy due to the vertical movement. The fluid

volume displacement at z=0 was specified for [V].

Integrated the total energy over the oscillation period [T], the average energy can be

obtained as:

= 0 (2.13)

[17]

( ) = + + (2.14)

2.3 FSI Consideration

Fluid-structure interactions (FSI) could be stable or oscillatory. The big idea in the FSI

analysis was determine damping terms and added mass in the mechanical system

oscillating in the liquid. The hydrodynamic added damping and damping mass is the

harmonic oscillation of the floating buoy generated waves which make dynamic pressure

around the buoy to volatile resulting in moments and forces exerting on that submerged

part of the floating buoy. In this thesis, the hydrodynamic added damping and added mass

in heave mode of the oscillation basis on the numerical method. Through the Fluid-

structure interaction (FSI) application mode in this MEMS module, the simulation and

modeling are carried out by using COMSOL Multiphysics. The (FSI) application

included a continuum application mode Stress-Strain, Two- Laminar, Tow-Phase Flow,

Phase Field (tpf), and Arbitrary Lagrangain-Eulerain (ALE) Physics Application.

2.3.1 Laminar, Two-Phase Flow, Phase Field (tpf) Physics Application

Using the Navier Stokes equation, equation of the continuity and moment transport

equation, the model equations of fluid are obtained to solve the velocity and pressure

[18]:

+ ( . ). = [ + { + ( ) }] + + + + (2.15)

This means that:

. = 0 (2.16)

Where:

: Fluid Density in [kg/ ]

: Fluid Velocity in [m/s]

[18]

: Dynamic Viscosity in [N.s/ ]

: Body Force in [N]

: Gravity Force in [N]

: Surface Tension Force in [N]

: Force due to External contribution.

The gravity force defined as:

= . (2.17)

Surface tension force is suitably considered as a body force which is defined as:

= . . (2.18)

But the force due to an external contribution to the free energy:

= (2.19)

Basis on the density of the fluid 1 ( ) and the fluid 2( ), the density of the fluid ( )

is calculated as:

= + ( ) . (2.20)

Also, the dynamic viscosity ( )of the fluid 1 ( ) and the fluid 2( ), the dynamic

= + ( + ) . (2.21)

Where:

: The volume fraction of fluid 2.

The volume fraction of fluid 2 can calculate as:

= min , 0 , 1 0 < < 1 (2.22)

[19]

According to [11], the = 0, so the Navier-Stokes equation for Laminar, Tow-Phase

Flow, and Phase Field obtained as:

+ ( . ). = [ + { + ( ) }] + + (2.23)

2.3.2 Stress-Strain Physics Application

In the Stress-Strain physics application assumed at the components of both of stresses

( , ) and strains ( , ) are zero. The global displacement (u, w) in z

direction can be solved. By assuming that the allowed loads are in x & z direction, the

equilibrium Stress-Strain equation can be [18]:

+ + + = 0 (2.24)

+ + + = 0 (2.25)

In order to the deformation components (u, v, w) which are associated with the strain-

displacement, the assumed of the small displacement in are:

= = = (2.26)

= + (2.27)

By Newton’s second law, the time-dependent equation to run transient analysis of the

buoy can obtain [18]:

= . = (2.28)

[20]

2.3.3 Arbitrary Lagrangain-Eulerain (ALE) Physics Application

Due to structural mechanics consideration and point of concentrating the studies,

following up the motion of the particle for a structure when dealing with meshing for the

numerical analysis, the mesh must follow the motion of the structure in the domain of

study.

A material coordinate system which know as, the Lagrangian method, which is a

technique used for a mesh movement such that each material particle coincided with the

node of the mesh (see Figure).

From fluid dynamics consideration, Lagrangian method was not applicable, since the

properties of the flow have apart relation of interest than particles, and where the

deformation was large too.

The Eulerian method was used since the mesh was fixed in the space (see Figure2.3), in

this method, the fluid motion and the flow properties changes of the shell be noticeable.

ALE method is used for handling the moving boundaries with a moving grid, and the

deforming geometry, where ALE algorithm allows a means of exploiting the duel

features (see Figure2.3).

The calculation was the dependent source for the movement of the mesh of the structure

which solved by Lagrangian algorithm.

Calculations for the fluid domain were solved by ALE algorithm; the mesh displacement

adjusted for a free displacement setting which was solved by Winslow smoothing method

[1], with constraints boundaries for solving the specified PDE. This significant that the

mesh has little deformation, where there was no presence of ill-shaped mesh elements.

[21]

Figure2.3 Lagrangian, Eulerian, and ALE description moving meshes [3]

2.4 Finite Element Analysis

Finite element analysis is a numerical solving way that resulting in an approximate

solution and percentage error. Finite element method use to solve the PDE (Partial

Deferential Equation) either by using several numerical integration techniques like

Range-Kutta, Midpoint theorem, Euler’s method, etc. Today, engineering conditions find

that it is important to get approximate numerical solutions to the problems instead of the

exact closed-form solution. Therefore using of the finite element method (FEM) is very

developing I the engineering analysis and they can expect this use to greatly in coming

year.

[22]

2.4.1 Finite Element Analysis Description

The main issue in the Finite element method while solving is first is to finding the

suitable weak forum equation. The second issue to specify the type of element that will

be used in the meshing if it is square or rectangular, etc... The result of the nodes in the

specified mesh gives the distribution of the desired solution [16].

2.4.2 Engineer Analysis Kinds

There is two considerations should be taken when analyzing any structure by modeling.

First, if the structure considered to behave in linearity that’s mean the material is not

enter the plastic region by having the ordinary parameters, where the other to behave is in

a non-linearity that’s mean the material deformed due to the capacity of stresses after it

overtaking the elastic region. Other type of analysis for a structure is to study its

resonance frequencies due to the vibrations occur of the body due to an impact, or

dynamic load, rotations, etc... It is important to know the resonance frequencies to avoid

failure in the system.

2.4.3 Practical Application

Due to unknown failure in the structure due to unspecified loads or stresses, the FEA give

the ability to predict that failure. Which shell reduces the cost of the manufacturing and

reduce the time of designing when dealing with well created modeling and well

estimating testing. [16]

2.5 Theoretical Model

A theoretical model is a general theory designed to explain the mechanical structures

characteristic and parameter effect to correct understanding of structure. In this part the

concept of the single degree of freedom [SDOF] system was present as the bases of the

theoretical model to model the dynamic structure of the body. The theoretical model of

[23]

the floating buoy was considered as a [SDOF] system corresponded to the added

damping, the buoyancy stiffness and added mass coefficient.

2.5.1 Single Degree of Freedom Model

The concept of a single degree of freedom (SDOF) is very important to develop the

dynamic structural analyses. Since it called Single degree due to the variation resulted on

one dimension. A single degree of freedom model is consisting of a mass, damper, and

spring as represent in figure 2.4.

f (t)

Figure2.4 SDOF system

A single degree of freedom system can be analyzed by the Newton’s second low equation

(2.29).

f (t) = m . a (2.29)

By the help of the free body diagram, figure 3.5, and the equation of motion (2.31)

express as:

f (t)

k (x) c (x)

Figure2.5 Free Body Diagram of SDOF System

M

[24]

( ) ( ) + ( ) = ( ) (2.30)

( ) + ( ) + ( ) = ( ) (2.31)

Where:

m: Mass of the body in (kg).

k: Spring stiffness,[ =( )

] in (N/m).

c: Coefficient of viscous damping, [ =( )

] in ( N.s/m).

x: The displacement of the mass from its equilibrium in (m).

f: The input force applied to the system in (N).

By taking the Laplace Transform of the equation 2.2, the system of equations has the general

solution of:

{x} = {X}.

{ } = s{X}. = s.x

{ } = {X}. = .

Where s: complex valued frequency

The system equation represented as:

( + + ). ( ) = ( ) (2.32)

By relating the Laplace transform of system to the Laplace transform system response,

the transform equation can be defined as:

H(s) = ( )

( )=( )

= / (2.33)

Where H(s) is known as a system transfer function of the SDOF system and that can be

written as the dynamic transfer function:

[25]

H(s) = ^

(2.34)

Where:

: The relative damping [ = ] in (rad/sec).

: The un-damping angular response frequency [ = ] in (rad/sec).

A simple way to obtain the Frequency Response Function of the SDOF system from

equation (2.33) which is represented in frequency domain as:

H(s) = + (2.35)

Where:

R: Residues for mode shape of the system.

: Poles of the system damping and natural frequency.

: Complex conjugate.

[26]

2.5.2 Mathematical Model

A mathematical model describes the significant characteristics by applying the physical

laws of the system to describe the dynamics for that system. The mathematical model of

the floating buoy was basic on SDOF mass-damper-spring system. The hydrodynamic

influence force which applied on the buoy can be founded from the integrating

hydrodynamic pressure. By solving velocity potential, the hydrodynamic pressure can

calculated over its wet surface. From the restoring force as state, the hydrostatic force can

obtained. By assumption about zero-forward- speed, the dynamic equation of motion

(2.36) cans shows as [13]:

[ + , ] + , + , = (2.36)

Where:

J, k: Signified of the mode no. [J, k= 1, 2, 3, 4, 5, 6] for sway, surge. Roll, heave, pitch

and yaw respectively.

: Expression for body inertia matrix.

, : Frequency-dependent added mass coefficient.

, : Damping coefficient.

, : Hydrostatic restoring force coefficients.

: Designated translator and angular displacements of the body.

: Exciting force for j= 1, 2, 3 and moment for j = 4, 5, 6 arose from the presence of the

incident wave elevation.

To present the heave mode of oscillation of the floating buoy, the equation (2.36) can

rewriting as following equation according to the figure 3.6:

[ + ( ) ] + ( ) + = ( ) (2.37)

[27]

Where:

( ): External exciting force considering to the hydrodynamic force in terms of

restoring force coefficient, damping coefficient and added mass.

Figure2.6 A SDOF model considering heaving

mode of oscillation of the floating buoy

[28]

3 Study Methodology

In this Study, the COMSOL Multiphysics software was used for analyzing the floating

buoys modeling numerically and solved the Finite Element Analysis (FEA) techniques.

3.1 Modeling

We will have two different domains when we used fluid structure interaction module

which called solid and fluid domain. In our model, we have two fluid domains air and

water and two solid domain for a cubic buoy which was considered for the modeling of

the floating buoy, and rectangular cubic buoy, this buoy would be excited by a harmonic

perturbation force of 700 [N] to bounce vertically up and down as a way to generate

ocean wave. The following below specified the model in detail.

3.1.1 Model Specification

The two solid floating buoys considered to be surrounding by two fluids the air, and part

of it is immersed in the water. The modeling was considered to be drawn in the

COMSOL Multiphysics by study the case in 2D space dimension. The air domain were

considered as a rectangle of width 6 [m] and height 1 [m] and the water domain is

considered as rectangle of width 6 [m] and height 2 [m] where the buoy domain is

considered as cubic with the 0.4 [m] side and wave generator domain is considers as

rectangular cubic with width 0.6[m] and height 0.4[m].

Figure 3.1 Dimension of the Model

[29]

3.1.2 Boundary conditions

For Arbitrary Lagrangian – Eulerian (ALE) branch, the moving mesh, solid, and water

domains boundary condition are discussed as well.

3.1.2.1 Moving Mesh Boundary Condition

In the modeling, the buoys that were represented as a solid domain were restricted to

move in the direction of the heave z-direction. Where the water and the air that

represented in the modeling as a two fluid domains where allowed to displace in all

direction after applying their boundary condition where the Moving Mesh ALE were

solved by the prescribed Winslow smoothing function.[1]

The fluid domain were formed by the Navier-Stokes equations, where the solution

appears in a on a freely moving deformed mesh, where the deformation were computed

by Winslow smoothing equation with respect to the initial shape. [1]

3.1.2.2 Solid Domain Boundary Condition

The two solid floating buoys were considered as a solid domain in our modeling, where

the Archimedes principle were used to figure out the density if the solid domain. The

principle tends that: any object immersed in a fluid partially or totally, is effected by a

force buoyed up, equal to the weight of the fluid displaced by the object [2]. The fluid

volumes that substituted from the submerged body were calculated as shown:

For the cubic buoy:

The mass of the buoy was denoted by and considered to be 20 kg and the density of

the water was denoted by .

× = ,

=20

1030= 0.01942 (3.2)

[30]

By apply Archimedes Principle to get the density of the buoy, where the density and the

volume of the buoy are represented by and respectively.

= 0.4 [ ] , =( . × )

.= 312.5 [ ] (3.3)

For the rectangular cubic buoy ( Wave generator):

The mass of the wave generator was denoted by and considered to be 40 kg and the

density of the water was denoted by .

× = ,

=40

1030= 0.0388 (3.4)

By apply Archimedes Principle to get the density of the wave generator, where the

density and the volume of the wave generator are represented by and

respectively.

= 0.4 0.6 = 0.096[ ] ,

=(0.0388 × 1030)

0.096= 416.67 [ ] (3.5)

[31]

Sitting

Description Cube Rectanguler cube

Width 0.4 0.6

Height 0.4 0.4

Weight 20 40

Density 312.5 416.67

Young's modulus 205e9 2750e6

Poisson's ratio 0.4 0.4

Table3.1 Solid domain parameter setting

The Boundary load in the solid domain is applied due to fluids surround it by the force

written as:

= 10 4 × (2 × × × ) [ ] (3.6)

= 10 4 × (2 × × × 2) [ ] (3.7)

:

2

:

:

[32]

3.1.2.3 Water and Air Domain Boundary Condition

In the fluid domain with respect to the water, it set to be incompressible constrained by

Navier-Stokes equation.

There were three walls, constrained as a no slip wall, the two edges boundaries of air and

water, and the bottom of the water, where No slip is the default boundary condition for the

stationary solid wall. This condition prescribes the velocity equal to zero u=0.

The walls of the two buoys considered as the Moving wetted wall for the solid domain

constrained with their velocity field, y component.

An outlet boundary was selected at the top of the modeling domain with a boundary

condition of Pressure P0=0 pa, with no viscous stress.

Figure 3.2.Fluid 1 and 2 for the floating buoy and wave generator

3.2 Model Meshing

In solving the model, the mesh shown below Fig 3.3 have been taken in consideration;

for the Time-dependence solver of a triangular type element where the mesh element

was 831 elements as given in the table 3.2 below:

[33]

Figure 3.3 Mesh 1

Mesh statistics

Property Value

Minimum element quality

0.5083

Average element quality

0.8957

Triangular elements

831

Edge elements 115

Vertex elements 18

Table3.2 Mesh statistics

[34]

3.3 Model Solver

For solving the modeling a solvers were chosen for the solution as explained below:

The Time Dependent were have the range from 0 to 10 sec with time step 0.001sec,

where the time stepping were depend on the Generalize alpha as the method for time

stepping where the steps taken were free unconstrained, where the Predictor was

considered as linear.

The solver with respect to the Direct where controlled in it the PARDISO method, where

a Factor in error estimate were 40 [18].

[35]

4 Result and Discussion4.1 Results

While solving the model in the Time-dependence solver, the mesh have been taken for a

triangular type meshing having a big size elements in most of the area while around the

buoy it become finer at the area of interest of the study .

Figure4.1 Mesh for Time-Dependence

The following figures shows the postprocessing of model in surface von Mises stress plots and contour and transient heave oscillation plots for both of buoy and wave generator:

Figure4.2 von Mises stress (N/m2) Surface Figure4.3 von Mises stress (N/m2) Surface (Wave Generator) (Buoy)

[36]

A unique excitation by point load [-700 N] where the minus sign tend to the vertical

downward direction of the load, as shown in figure 4.2 had been applied. The movement of

wave generator buoy after this unique excitation will be generates the wave in the domain of

this study. And the two buoys will be bouncing with damping.

The buoy start damped bouncing due to the wave generated, as figure 4.3 shown. By this

bouncing, the displacement of the buoy was plotted in figure 4.7 to trace the frequencies of

the buoy.

Figure4.4 Volume fraction of fluid 1 Figure4.5 Phase field vs Time

The volume fraction of the fluid was been calculated by this equation:

[1-min (max (0.5*(1+phipf), 0), 1)], where phipf is the phase field variable which is unitless

varies by means of time in the sinusoidal damped shape as shown in figure 4.5 where the

equation show the maximum and minimum of the phase field to plot the volume fraction of

unit 1 as figure 4.4 shown.

[37]

Figure4.6 Velocity magnitude (m/s)Contour: Volume fraction of fluid 1

The velocity magnitude of the fluid was been calculated by[ ( + )], where and

of the velocity field component of x and y directions, as figure 4.6 shown.

Figure4.7 Viscosity (pa*s) Figure4.8 Viscosity Distribution

To ensure that the modeling of the tank behave as the real ocean where the wave in the ocean

don’t reflect an equation of the viscosity have been implemented in the COMSOL to prevent

the reflection of the wave as shown in figure 4.7 and figure 4.8. This equation will raise the

[38]

viscosity of the water at the edges and keep it normally elsewhere to guarantee no reflection

of the wave and the only wave that will excite the buoy is the wave generated by the wave

generator as shown below.

Since in the real life the buoy is bouncing in an infinite surround that’s mean the wave

will generated will not stuck and reflected back, they will keep moving until the damped

to zero when reach edges. In order to perform that the viscosity at the edges should be

high to damp the waves and won’t let them reflected back. The equation of the viscosity

was written as:

Vd*exp1*exp2

where exp1 = 2.7183^(1e-7/x)

exp2 = 2.7183^(1e-7/abs(6-x))

Vd= 8.9e-4[Pa*s]

At the two side edges we consider it as a no slip wall and the upper edges as an outlet

since the tank its open to the air in real life.

Figure4.9 Displacement of the Wave Generator

This figure 4.9 shows the displacement of the wave generator after excitation by the load, it’s

moving in a damped sinusoidal wave shape bouncing up and down.

[39]

Figure4.10 Displacement of the buoy

Figure4.11 End of the fisrt peorid

This figure 4.10 shows the displacement of the buoy after excitation by the wave, it’s moving

in a damped sinusoidal wave shape bouncing up and down.

[40]

Since this study is held in one dimension in vertical direction, the displacement of the buoy

up and down was taken in consideration a single degree of equation was control the study to

get the frequency of the system.

25 + 25 + 490.5 = 0

Where:

1- The Mass of buoy= 20 kg and the Added mass=5kg, the total mass= 25 kg

2- Damping: = + =1 (1/s) x 25 (Kg) =25 N.s/m

where

Beta : Rayleigh damping

Alfa : Mass damping parameter

3- Stiffness damping parameter:

=( )= 490.5 /

Figure 4.12 SDOF

By the help of the free body diagram, the equation of motion express as:

Where: m: Mass of the body (kg).

k: Spring stiffness (N/m) where, .

c: Coefficient of viscous damping( N.s/m) where

z: The displacement of the mass from its equilibrium in (m).

[41]

The Frequency of the buoy is calculated by the reciprocal of the time taken to finish the

first cycle from the periodic displacement graph shown above.

=1

1.2340.81

The main issue in this physics is to specify the viscosity of the fluid, specify the walls for

the boundary condition, and the volume force.

Results:

The frequency where equal to 0.81 Hz

4.2 Parameters Estimation

The parameters estimation was handled by means of the spectrum analysis, where the

heave oscillation was evaluate by the discrete Fourier transform (DFT) by means of

computational algorithms fast Fourier transform (FFT). The spectrum of the heave

response was seized by the weight of the buoy, the spectrum of the input force, as a result

of the frequency response receptance. The added mass was evaluated by log-log plot of

the receptance of the slope for high frequency. The added damping is then calculated

through the damping ratio definition.

[42]

5. Conclusion

This model was handled under SDOF, simplified to mass, spring and damper, the model

was describe the dynamics motion for a semi-submerged floating buoy, where an

additional terms were added like added mass and added damper.

Where these variables were depend on the frequency of oscillating and their modes, the

geometry of the floating buoy, and water-depth.

The Model was considered as the buoy was affected by a load on the buoy and affected

by the wave created in the fluid domain to oscillate the buoy and the spongy wall of the

tank to prevent the reflection of the wave after inserting the equation of the viscosity of

the water.

The heave displacement of the model under the time-dependent was obtained by means

of FEM, solving the coupled partial differential equations of ALE, stress-strain equations,

and Navier-stoke equations.

By the result of the displacement of the buoy the frequency was extracted from the graph

of the displacement.

The numerical model SDOF was compared by the results from the previous work [9, 10].

The results were within the acceptable range.

Future work

A suggestion for further work is extending of this work can be by researching about

considered the heavy wind and the strong waves and their effect on the oscillating

floating buoy. Researchers could confirm the results with any other numerical software.

Also may perform two or multi degree of freedom for this study. Held this study in salty

water instead of fresh water and observes the bouncing and the frequency. Perform other

way of generating wave like moving wall.

[43]

6. Reference1. ALE Fluid-Structure Interaction,

http://www.comsol.com/shared/images/enews/2011/feb/models_mems_fluid_structure_in

teraction.pdf?utm_source=PDF+Download&utm_campaign=us_enews_feb11&utm_med

ium=e-News&utm_content=3

2. Acott, Chris (1999). "The diving "Law-ers": A brief resume of their

lives.". South Pacific Underwater Medicine Society journal 29 (1) ISSN 0813-

1988. OCLC 16986801.

3. A three-dimensional parallel adaptive mesh refinement method for fluid

structure Interaction, Institution: University of Virginia, Author: Jean-Noel Pederzani,

Year: 2009, EAN: 9781109676662, Pages: 106.

4. Hamid Sarlak, M.S. Seif and M. Abbaspour, (2010), Experimental

Investigation of Offshore Wave Buoy Performance, Journal of Marine Engineering.

5. Conca, C., Osses, A., and Planchard, J, (1996), Added Mass and Damping

in Fluid-Structure Interaction, Computer Methods in Applied Mechanics and

Engineering, Elsevier Science S.A.

6. Aquelet, N. and Souli, M, (2003), Damping Effect in Fluid-Structure

Interaction: Application to Slamming Problem, Pp. 175-184, ASME 2003 Pressure

Vessels and Piping Conference July 20-24, 2003, Cleveland, Ohio, USA, ISBN: 0-7918-

4155-3

7. WU, G.X. (1998), Wave Radiation and Diffraction by a Submerged Sphere

in a Channel, Vol. 51, Pp. 647-666, Oxford Journals, Oxford University Press.

8. Sharan, K. Shailendra. (1986), Modelling of Radiation Damping in Fluid by

Finite Elements, Vol. 23, Pp. 945-957, International Journal for Numerical Methods in

Engineering.

9. Hanif, Muhammad A., Oad, Rajev K., Patel, Rakeshhai, (2011), fluid

structure interaction analysis of planar buoy, Master’s degree thesis, Blekinge Institute

of Technology, Sweden, ISRN: BTH-AMT-EX--2011/D-08--SE.

[44]

10. Ramadan, Mohamed S., Saichua, Sorawan, (2010), fluid-structure

interaction analysis for hydrodynamic parameters in semi-submerged axisymmetric buoy,

Master’s degree thesis, Blekinge Institute of Technology, Sweden, ISRN: BTH-AMT-

EX--2010/D-08--SE.

11. Esedig, Ahmedelrayah M., Osman, Hazzaa A., and Fadaee, Mohammad,

(2009), Modeling of buoyancy and motion of a submerged body, Master’s degree thesis,

Blekinge Institute of Technology, Sweden, ISRN: BTHAMT-EX--2009/D-03--SE.

12. Hazel, L. Andrew, Heil, Matthias, (2006), Oomph-lib-An ObjectOriented

Multi-Physics Finite-Element Library, Springer-Verlag Berlin Heidelberg, The

Netherlands, ISBN-I3 978-3-540-34595-4.

13. Clauss, F. Günther, Schmittner, E. Christian, and Stutz, Katja, (2003),

Freak wave impact on semisubmersibles - Time-domain analysis of motions and forces,,

Research project MAXWAVE, Institute of Land and Sea Transport Systems Technical

University of Berlin, Germany. 57

14. Faltinsen, O.M., (1990), Sea loads on ships and offshore structures,

Cambridge University Press, United Kingdom, ISBN: 0-0521458706

15. J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodriguez-Ferran, (2004),

Arbitrary Lagrangian-Eulerian methods, Encyclopedia of Computational Mechanics.

16. Finite element analysis, site:

http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/num/widas/ history.html,

Accessed on 13th August, 2011

17. Conca, C., Osses, A., Planchard, J., (1997), Added mass and damping in

fluid-structure interaction, Vol. 146, Pp. 387-405, Computer Methods in applied

mechanics and engineering, Elsevier Science B.V.

18. COMSOL Multiphysics 4.0a, (June 2010), COMSOL Multiphysics

Reference Guide 4.0a, Part number: CM020005

19. Martinelli, Luca and Ruol, Piero, (2006), 2D model of floating breakwater

dynamics under linear and nonlinear waves, 2nd COMSOL Conference, 14 Nov.,

Milano.

[45]

20. Falnes, Johannes. (2002). Ocean waves and oscillating system, 287 pp.,

Cambridge University Press, United Kingdom. ISBN: 0-52178211-2.

[46]

Appendix 1

1. Selected Space Dimension

> 2D

2. Add Physics

> Moving Mesh (ale) > Solid Mechanics (solid)> Laminar Two-Phase Flow, Moving Mesh

(tpfmm)

3. Selected Study Type

>Time Dependent > Finish

4. Model 1(mod1)

A. Definitions

i. Geometry

Geometry statisticsProperty Value

Space dimension 2Number of domains 6

Number of boundaries 23

ii. Water (r1)iii.

Settings Name Value

Position {0, 0} Width 6Height 2Size {6, 2}

[47]

iv. Air (r2)

Settings Name Value

Position {0, 2} y 2Width 6Height 1Size {6, 1}

v. Buoy (r3)

vi. Wave Generator (r4)

Settings Name Value

Position {4.5, 2} x 4.5y 2Base Center Side length 0.4Side length 0.4Size {0.4,0.4}

Settings Name Value

Position {0.6, 2} x 0.6y 2Base Center Side length 0.6Side length 0.4Size {0.6,0.4}

[48]

Moving Mesh (ale)

Moving Mesh selection

Settings

Description Value

Geometry shape order 1

Mesh smoothing type Winslow

[49]

vii. Fixed Mesh 1

1. Selection

Fixed Mesh 1

viii. Prescribed Mesh Displacement 1

1. Settings Settings

Description Value

Prescribed # displacement {1, 0}

2. Selection

Prescribed Mesh Displacement 1

[50]

ix. Prescribed Deformation 1

1. Settings Settings

Description Value

Prescribed mesh displacement {0, v}

2. Selection

Prescribed Deformation 1

x. Free Deformation 1

1. Selection

Free Deformation 1

[51]

xi. Prescribed Mesh Displacement 4

1. Selection

Prescribed Mesh Displacement 4

xii. Prescribed Mesh Displacement 2

1. SettingsSettings

Description Value

Prescribed mesh displacement {0, v}

2. Selection

Prescribed Mesh Displacement 2

[52]

B. Solid Mechanics (solid)

C.

Solid Mechanics

Settings

Description Value

Show equation assuming std1/time

i. Linear Elastic Material Model 1

1. Selection

Linear Elastic Material Model 1

[53]

Settings

Description Value

Young's modulus User defined

Young's modulus 205e9

Poisson's ratio User defined

Poisson's ratio 0.4

Density User defined

Density DensityBouy

Include geometric nonlinearity 1

ii. Free 1

1. Selection

Free 1

[54]

iii. Initial Values 1

1. Selection

Initial Values 1

iv. Linear Elastic Material Model 2

1. Settings Settings

Description Value

Poisson's ratio User defined

Poisson's ratio 0.4

Density User defined

Density DensityBouy

[55]

2. Selection

Linear Elastic Material Model 2

v. Body Load 1

1. Settings Settings

Description Value

Body Load {0, -BodyLoadB, 0}

2. Selection

Body Load 1

[56]

vi. Boundary Load 1

1. Settings

Settings

Description Value

Total Force {0, -0.1*10e4*sin(2*pi/21*t*v), 0}

2. Selection

Boundary Load 1

Roller 1

1. Settings

Description Value

Constraint type Unidirectional

Use weak constraints 1

[57]

2. Selection

Roller 1

2.1.1. Added Mass 1

1. Settings

Settings

Description Value

Total mass {0, 5, 0}

2.Selection

Added Mass 1

[58]

D. Laminar Two-Phase Flow, Phase Field (tpf)

Laminar Two-Phase Flow, Phase Field

Settings

Description Value

Multiphase flow model Two-phase flow, phase field

Show equation assuming std1/time

i. Fluid Properties 1

1. Settings

Settings

Description Value

Density User defined

Dynamic viscosity Vs

Surface tension coefficient User defined

[59]

Parameter controlling interface thickness 0.07

1. Selection

Fluid Properties 1

i. Fluid Properties 3

1. Settings Settings

Description Value

Density User defined

Dynamic viscosity Vs

Surface tension coefficient User defined

Parameter controlling interface thickness 0.07

2. Selection

Fluid Properties 3

[60]

ii. Volume Force 1

1. Settings

Settings

Description Value

Volume force {0, -9.81*if(phipf<0,1.25,1030), 0}

2. Selection

Volume Force 1

iii. Volume Force 2

1. SettingsSettings

Description Value

Volume force {0, -9.81*if(phipf<0,1.25,1030), 0}

[61]

2. Selection

Volume Force 2

iv. Initial Values 2

1. Settings

Settings

Description Value

Fluid initially in domain Fluid 2

2. Selection

Initial Values 2

v. Wall 2

1. SettingsSettings

[62]

Description Value

Boundary condition Moving wetted wall

Velocity of moving wall {0, vt, 0}

2. Selection

Wall 2

[63]

vi. Wall 3

1. SettingsSettings

Description Value

Boundary condition Moving wetted wall

Velocity of moving wall {0, v2t, 0}

2. Selection

Wall 3

vii. Wall 4

1. Selection

Wall 4

[64]

viii. Wall 5

1. Selection

Wall 5

ix. Outlet 1

1. Settings Settings

Description Value

Boundary condition Pressure

2. Selection

Outlet 1

[65]

b. Meshes

i. Mesh 1

Mesh statistics

Property Value

Minimum element quality 0.5083

Average element quality 0.8957

Triangular elements 831

Edge elements 115

Vertex elements 18

ii. Selection

Outlet 1

[66]

1. Size (size) Settings

Name Value

Calibrate for Fluid dynamics

Maximum element size 0.6

Minimum element size 0.012

Resolution of curvature 0.4

Maximum element growth rate 1.4

Predefined size Coarse

c. Study 1

i. Time Dependent

Times: range(0,0.001,10)

Mesh selection

Geometry Mesh

Geometry 1 (geom1) Mesh1

Physics selection

Physics interface Discretization

Solid Mechanics (solid) physics

Solid Mechanics 2 (solid2) physics

Moving Mesh (ale) physics

Laminar Two-Phase Flow, Phase Field (tpf) physics

School of Engineering, Department of Mechanical Engineering Blekinge Institute of Technology SE-371 79 Karlskrona, SWEDEN

Telephone: E-mail:

+46 455-38 50 00 info@bth.se